Journal of Geometry and Physics 61 (2011) 1928–1931 EMBEDDING ALMOST-COMPLEX MANIFOLDS IN ALMOST-COMPLEX EUCLIDEAN SPACES ANTONIO J. DI SCALA AND DANIELE ZUDDAS Abstract. We show that any compact almost-complex manifold (M; J) of complex dimension m can be pseudo-holomorphically embedded in R6m equipped with a suitable almost-complex structure Je. Keywords: embedding, almost-complex structure, manifold, pseudo-holomorphic embedding. 1. Introduction An almost-complex structure on a 2n-dimensional smooth manifold M is a tensor J 2 End(TM) such that J 2 = − id. If M is oriented we say that J is positive if the orientation induced by J on M agrees with the given one. An almost-complex structure is called integrable if it is induced by a holomorphic atlas. In dimension two any almost-complex structure is in- tegrable, while in higher dimension this is far from true. A smooth map f : N ! M between two almost-complex manifolds (N; J 0), (M; J) is called pseudo-holomorphic if J ◦ T f = T f ◦ J 0, where T f : TN ! TM is the tangent map of f. When the map f is an embedding, (N; J 0) is said to be an almost-complex submanifold of (M; J). In this case we can identify N with its image f(N) ⊂ M and the almost-complex structure J 0 with the restriction of J to TN ∼= T (f(N)) ⊂ TM. 2n 2n ∼ n If we equip R with the canonical complex structure, that is to say R = C , then it does not admit any compact complex submanifold (by the maximum principle). Thus, it is a very natural problem to ascertain if it is possible to find compact complex manifolds pseudo-holomorphically embedded in R2n equipped with an integrable or non-integrable almost-complex structure. In [2] Calabi and Eckmann constructed the first examples of compact, simply connected com- 2p+1 2q+1 plex manifolds Mp;q which are not algebraic. Topologically Mp;q is the product S × S . Then by deleting a point on each factor one obtains a complex structure J on R2p+2q+2. In section 5 of [2] it was shown that when p; q > 1 there exists a complex torus as a complex submanifold of (R2p+2q+2;J) [2, p. 499]. It follows that the Calabi-Eckmann complex structure J on R2n cannot be tamed by any symplectic form and in particular cannot be K¨ahler.Cal- abi and Eckmann also observed that the only holomorphic functions on (R2p+2q+2;J) are the constants answering negatively to a question raised by Bochner about the uniformization of complex structures on R2n. In [1] Bryant constructed pseudo-holomorphic non-constant maps ' : M 2 ! S6 for any compact Riemann surface M 2, where S6 is equipped with the almost- complex structure induced by the octonion multiplication. These maps realize compact Riemann surfaces as pseudo-holomorphic singular curves in S6. In [3] it was shown that any almost-complex torus Tn = R2n=Λ can be pseudo-holomor- 4n phically embedded into (R ;JΛ) for a suitable almost-complex structure JΛ. It follows that 2000 Mathematics Subject Classification. Primary 32Q60 ; Secondary 32Q65. This work was supported by the Project M.I.U.R. PRIN 05 \Riemannian Metrics and Differentiable Mani- folds" and by G.N.S.A.G.A. of I.N.d.A.M. The second author was supported by Regione Autonoma della Sardegna with funds from PO Sardegna FSE 2007{2013 and L.R. 7/2007 \Promotion of scientific research and technological innovation in Sardinia". 1 DOI: 10.1016/j.geomphys.2011.05.002 2 ANTONIO J. DI SCALA AND DANIELE ZUDDAS any compact Riemann surface can be realized as a pseudo-holomorphic curve of some (R2n;J), where J is a suitable almost-complex structure. In this paper we prove the following general theorem. Theorem 1. Any compact almost-complex manifold (M; J) of real dimension 2m can be pseudo- holomorphically embedded in (R6m; Je) for a suitable positive almost-complex structure Je. In particular, any compact Riemann surface can be realized as a pseudo-holomorphic curve in (R6; Je). In [3] was shown that the torus is the only compact Riemann surface that can be pseudo-holomorphically embedded in (R4; Je) for some Je. 2. Preliminaries The space of positive linear complex structures on R2n is diffeomorphic to the homogeneous space eJ(n) = GL+(2n; R)=GL(n; C) and is homotopy equivalent to J(n) = SO(2n)=U(n). So, an almost-complex structure J on R2n can be regarded as a smooth map J : R2n ! eJ(n). Lemma 2. Let M ⊂ R2n be a closed submanifold and let J : M ! eJ(n) be a smooth map. Then there exists a smooth extension Je : R2n ! eJ(n) if and only if J is homotopic to a constant. Proof. The `only if' part follows immediately from the fact that R2n is contractible. Let us prove the `if' part. Consider a smooth homotopy H : M × [0; 1] ! eJ(2n) such that H0(x) = J0 for all x 2 M, and H1 = J where Ht(x) = H(x; t) and J0 2 eJ(n). We can extend H 2n 2n 2n to R ×f0g ⊂ R ×[0; 1] by setting H(x; 0) = J0 for any x 2 R . By the homotopy extension property [4, Chapter 0] there exists He : R2n × [0; 1] ! eJ(n) which extends H. We conclude the proof by setting Je = He1. Let (M; J) be an almost-complex manifold. The strategy to prove Theorem 1 will be to choose an arbitrary embedding f : M,! R6m, which exists for the weak Whitney embedding theorem, and to show that J extends to the pullback f ∗(T R6m) and this extension is null-homotopic. Consider the standard filtration SO(1) ⊂ SO(2) ⊂ · · · . Since SO(n−1) contains the (n−2)- skeleton of SO(n) (because the standard fibration SO(n) ! Sn−1) it follows that the k-skeleton of SO(n) is contained on SO(k + 1) for 0 ≤ k ≤ n − 2. Since SO(n) ⊂ U(n) it follows that U(n) contains the (n − 1)-skeleton of SO(2n) for n ≥ 1. Then the homomorphism induced by the inclusion i∗ : πj(U(n)) ! πj(SO(2n)) is an isomor- phism for j ≤ n − 2 and is an epimorphism for j = n − 1. From the homotopy exact sequence of the fibre bundle SO(2n) ! J(n) given by the projection ∼ ∼ map it follows that πj(eJ(n)) = πj(J(n)) = 0 for j ≤ n − 1. ∼ Definition 3. A space X is said to be n-connected if πj(X) = 0 for all j ≤ n. In particular, 0-connected means path-connected. From the above considerations we have that eJ(n) is (n − 1)-connected. The following propo- sition is well-known in the theory of CW-complexes. Proposition 4. If X is n-connected then any map Y ! X defined on a CW-complex Y of dimension ≤ n is homotopic to a constant. Also the following proposition is standard, and we give only the idea of the proof. Proposition 5. Let ξ : E ! M be an oriented real vector bundle of rank 2k over an m-manifold M. If k ≥ m then ξ admits a positive complex structure. EMBEDDING ALMOST-COMPLEX MANIFOLDS 3 Proof. Consider the bundle ξJ : eJ(E) ! M with fibre eJ(k) induced by ξ. Namely, for any p 2 M the fibre of ξJ over p is the space of positive linear complex structures on ξ−1(p). Since eJ(k) is (k − 1)-connected, it follows that ξJ admits a section if k ≥ m, see [7, Part III]. This section is a positive complex structure on ξ. N Let f : M ! R be an immersion. The normal bundle νf (M) is, as usual, the orthogonal complement of TM in f ∗(T RN ), that is to say: ∗ N f (T R ) = TM ⊕ νf (M): If M is oriented then the normal bundle can be equipped with a canonical orientation, namely that which makes the splitting of f ∗(T RN ) into a Whitney sum of oriented fibre bundles, where RN is considered with the standard orientation. 3. Proof of the main results Theorem 6. Let M ⊂ R2n be a submanifold of even dimension endowed with an almost- complex structure J. If the normal bundle of M in R2n admits a positive complex structure with respect to the canonical orientation, then for any k ≥ max(0; dimR M − n + 1) there exists an almost-complex structure Je on R2n × R2k such that M × f0g ⊂ R2n × R2k is an almost-complex submanifold. Proof. Let us choose a positive complex structure on the normal bundle of M. Then by taking the Whitney sum with the almost-complex structure on M we get a complex structure on 2n (T R )jM . So we obtain a smooth map J : M ! eJ(n). In view of Lemma 2 our target is to get a J null-homotopic. This is so if dimR M ≤ n − 1 because eJ(n) is (n − 1)-connected and Proposition 4. 2n 2k 2k If dimR M > n − 1 we take the product R × R , where R is endowed with the standard complex structure, and we embed M as M × f0g. We get a complex structure on the normal 2n 2k bundle of M in R ×R in the obvious way. So we obtain a map Jk : M ! eJ(n+k). It follows 2n 2k that Jk is homotopic to a constant if k ≥ dimR M − n + 1.